# lec10

## Half-adder

This will be the building block for adding bit-strings together later on however, for now we are going to just add two singular bits.
To accomplish this we'll build a half adder.

This means our logic circuit must adhere to the following logic table.
If both inputs are 0 then our result is 0 and we don't have to carry anything out.
If only one input A/B is 1 then our result will clearly be 1 and our carry will be 0. 
Finally if both inputs are 0 then since we can't fit 2 in a single bit it means we have to carry-out a 1, and our result will be 0.

With all of this in mind we have a table to guide how we will implement our logic circuit.
I __highly__ suggest that you try to build a logic circuit on your own first as most of the content is best learned through practice.

| A | B | Carry-out | Result |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 1 |
| 1 | 0 | 0 | 1 |
| 1 | 1 | 1 | 0 |

![fig0](../img/fig0lec10.png)

## Full Adder

If we only want to add single-bit's then a half-adder works fine but if we want to add multiple bits say `1011 + 0010` then we need to consider that we will likely have to chain these together.
The full-adder has 1 main difference from the half-adder, it has 3 inputs, 2 main inputs and 1 input for the carry bit.
The carry bit will propagate along the operation now if we chain these together, _just like real addition!_

![fig1](../img/fig1lec10.png)

With this we can start chaining together multiple Full-adders we can start adding multiple bits at the same time since the carry now propagates along the chain.

## Ripple Adders

An N-bit adder is really just made up of Full adders chained together.
Each adder is chained to the next by the carry-out line which then acts as the next adder's carry-in line.
If we have say a 4-bit ripple adder, then each bit in the bit strings will go to a different adder. 
For now the initial carry in bit will be fed a 0 everytime.

![fig2](../img/fig2lec10.png)

Here we see that our 4-bit input A & B have the values `1111` & `0000` respectively. 
The 0th bit goes to Adder0, the 1th goes to Adder1, and so on.
You should have also noticed that each one also takes a carry-in, which for now is 0 but it does mean we have to let the comutation happen one adder at a time.
None of the sequential adders can do anything if the previous have done anything yet either.
At the end of it all we get some bit results, and a carry-out. 
We can then simply reassemble the results back into one bit-string to assemble our final result.